Finite Group Databasebeta

← C3×#29841 #10648448

Subgroups — C3×#29841 #10648448

Order 768.

754 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#475C211yesmaximal, characteristic, complemented
2C2×C3×#464C211yesmaximal, characteristic
3C3×#2640C211yesmaximal, characteristic
4C3×#2320C211yesmaximal, characteristic, complemented
5C3×#2066C212yesmaximal, complemented
6C3×#2065C212yesmaximal
7C3×#2056C212yesmaximal, complemented
8C3×#2055C212yesmaximal
9C3×#1987C211yesmaximal, characteristic, complemented
10C3×#1983C211yesmaximal, characteristic
11C3×#1843C211yesmaximal, characteristic, complemented
12#29841C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2211yescharacteristic
14C22×C3×SD16C2211yescharacteristic
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#124C2211yescharacteristic
17C2×C3×C4×Q8C2211yescharacteristic
18C3×#475—24no—
19C3×#464—24no—
20C2×C3×#106C2212yes—
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×C4⋊C8C2212yes—
24C2×C3×C4⋊C8C2212yes—
25C2×C3×#103C2212yes—
26C2×C3×#103C2211yescharacteristic
27C2×C3×#103C2212yes—
28C2×C3×#102C2211yescharacteristic
29C2×C3×#102C2212yes—
30C3×#397—22no—
31C3×#393—21no—
32C3×#393—21no—
33C3×#391C2212yes—
34C3×#391C2212yes—
35C3×#390C2212yes—
36C3×#389C2212yes—
37C3×#387—22no—
38C3×#383—21no—
39C3×#382—21no—
40C3×C4×C4⋊C4C2211yescharacteristic
41C3×#376C2211yescharacteristic
42C3×#340C2212yes—
43C3×#340C2212yes—
44C2×#475C2×C311yescharacteristic, complemented
45C2×#464C2×C311yescharacteristic
46#2640C2×C311yescharacteristic
47#2320C2×C311yescharacteristic, complemented
48#2066C2×C312yescomplemented
49#2065C2×C312yes—
50#2056C2×C312yescomplemented
51#2055C2×C312yes—
52#1987C2×C311yescharacteristic, complemented
53#1983C2×C311yescharacteristic
54#1843C2×C311yescharacteristic, complemented
55C22×C3×Q8—21no—
56C22×C3×Q8C2311yescharacteristic
57C22×C3×Q8C2311yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C2×C3×Q16—22no—
60C2×C3×Q16—42no—
61C2×C3×SD16—22no—
62C2×C3×SD16—42no—
63C22×C3×C8C2311yescharacteristic
64C22×C3×C8C2311yescharacteristic
65C22×C3×C8C2312yes—
66C3×C4⋊Q8—22no—
67C3×C4⋊Q8—22no—
68C3×#124—22no—
69C3×#124—22no—
70C3×C4×Q8—22no—
71C3×C4×Q8D414yes—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C22⋊C4—21no—
79C2×C3×C22⋊C4—21no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C42—21no—
83C2×C3×C42—21no—
84C2×C3×C42C2312yes—
85C2×C3×C42C2311yescharacteristic
86C3×#106—24no—
87C3×C4⋊C8—22no—
88C3×C4⋊C8—22no—
89C3×C4⋊C8—42no—
90C3×C4⋊C8—42no—
91C3×#103—24no—
92C3×#103—22no—
93C3×#103—42no—
94C3×#102—22no—
95C3×#102—42no—
96C3×#95—21no—
97C3×#95—22no—
98C3×#95—22no—
99C3×#95—22no—
100C3×#95—22no—
101C3×#95—22no—
102C3×#95D411yescharacteristic
103C3×#95D411yescharacteristic
104C3×#95D412yes—
105C22×Q16C22×C311yescharacteristic
106C22×SD16C22×C311yescharacteristic
107C2×C4⋊Q8C22×C311yescharacteristic
108C2×#124C22×C311yescharacteristic
109C2×C4×Q8C22×C311yescharacteristic
110#475—24no—
111#464—24no—
112C2×#106C22×C312yes—
113C2×C4⋊C8C22×C311yescharacteristic
114C2×C4⋊C8C22×C311yescharacteristic
115C2×C4⋊C8C22×C312yes—
116C2×C4⋊C8C22×C312yes—
117C2×#103C22×C312yes—
118C2×#103C22×C311yescharacteristic
119C2×#103C22×C312yes—
120C2×#102C22×C311yescharacteristic
121C2×#102C22×C312yes—
122#397—22no—
123#393—21no—
124#393—21no—
125#391C22×C312yes—
126#391C22×C312yes—
127#390C22×C312yes—
128#389C22×C312yes—
129#387—22no—
130#383—21no—
131#382—21no—
132C4×C4⋊C4C22×C311yescharacteristic
133#376C22×C311yescharacteristic
134#340C22×C312yes—
135#340C22×C312yes—
136C24×C3—21no—
137C2×C3×Q8—21no—
138C2×C3×Q8—21no—
139C2×C3×Q8—21no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×Q8—22no—
143C2×C3×Q8—22no—
144C2×C3×Q8—42no—
145C2×C3×Q8—82no—
146C2×C3×Q8C2×D412yes—
147C2×C3×D4—21no—
148C2×C3×D4—21no—
149C2×C3×D4—22no—
150C2×C3×D4—22no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2411yescharacteristic
161C3×Q16—44no—
162C3×SD16—44no—
163C2×C3×C8—21no—
164C2×C3×C8—22no—
165C2×C3×C8—21no—
166C2×C3×C8—22no—
167C2×C3×C8—22no—
168C2×C3×C8—22no—
169C2×C3×C8—22no—
170C3×C4⋊C4—24no—
171C3×C4⋊C4—21no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—42no—
177C3×C4⋊C4—82no—
178C3×C4⋊C4C2×D412yes—
179C3×C22⋊C4—42no—
180C3×C22⋊C4—42no—
181C3×C22⋊C4—82no—
182C3×C22⋊C4—82no—
183C3×C42—24no—
184C3×C42—22no—
185C3×C42—22no—
186C3×C42—81no—
187C3×C42C2×D412yes—
188C3×C42#4112yes—
189C22×Q8—21no—
190C22×Q8C23×C311yescharacteristic
191C22×Q8C23×C311yescharacteristic
192C22×D4C23×C311yescharacteristic
193C2×Q16—22no—
194C2×Q16—42no—
195C2×SD16—22no—
196C2×SD16—42no—
197C22×C8C23×C311yescharacteristic
198C22×C8C23×C311yescharacteristic
199C22×C8C23×C312yes—
200C4⋊Q8—22no—
201C4⋊Q8—22no—
202#124—22no—
203#124—22no—
204C4×Q8—22no—
205C4×Q8C3×D414yes—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4C23×C312yes—
210C2×C4⋊C4C23×C312yes—
211C2×C4⋊C4C23×C311yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42C23×C312yes—
219C2×C42C23×C311yescharacteristic
220#106—24no—
221C4⋊C8—22no—
222C4⋊C8—22no—
223C4⋊C8—42no—
224C4⋊C8—42no—
225#103—24no—
226#103—22no—
227#103—42no—
228#102—22no—
229#102—42no—
230#95—21no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95C3×D411yescharacteristic
237#95C3×D411yescharacteristic
238#95C3×D412yes—
239C23×C3—41no—
240C23×C3—42no—
241C23×C3—42no—
242C23×C3—41no—
243C23×C3—41no—
244C23×C3C22×D411yescharacteristic
245C3×Q8—22no—
246C3×Q8—22no—
247C3×Q8—42no—
248C3×Q8—41no—
249C3×Q8—41no—
250C3×Q8—42no—
251C3×Q8—84no—
252C3×D4—41no—
253C3×D4—41no—
254C3×D4—42no—
255C2×C3×C4—21no—
256C2×C3×C4—22no—
257C2×C3×C4—21no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—42no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4#12112yes—
278C2×C3×C4#12112yes—
279C2×C3×C4C22×D411yescharacteristic
280C2×C3×C4C2×#4111yescharacteristic
281C2×C3×C4H(5,𝔽2)12yes—
282C2×C3×C42−1+412yes—
283C3×C8—42no—
284C3×C8—41no—
285C3×C8—41no—
286C24—21no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—42no—
295C2×Q8—82no—
296C2×Q8C2×C3×D412yes—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—22no—
300C2×D4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4C2×C3×D412yes—
307C22×C4C2×C3×D412yes—
308C22×C4C2×C3×D411yescharacteristic
309C22×C4C2×C3×D411yescharacteristic
310C22×C4C24×C311yescharacteristic
311Q16—44no—
312SD16—44no—
313C2×C8—21no—
314C2×C8—22no—
315C2×C8—21no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C4⋊C4—24no—
321C4⋊C4—21no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—42no—
327C4⋊C4—82no—
328C4⋊C4C2×C3×D412yes—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—82no—
332C22⋊C4—82no—
333C42—24no—
334C42—22no—
335C42—22no—
336C42—81no—
337C42C2×C3×D412yes—
338C42C3×#4112yes—
339C22×C3—81no—
340C22×C3—82no—
341C22×C3—82no—
342C22×C3—81no—
343C22×C3—81no—
344C22×C3C2×#12111yescharacteristic
345C22×C3#53712yes—
346C22×C3#57712yes—
347C22×C3#57711yescharacteristic
348C22×C3#57811yescharacteristic
349C3×C4—22no—
350C3×C4—22no—
351C3×C4—42no—
352C3×C4—42no—
353C3×C4—42no—
354C3×C4—42no—
355C3×C4—81no—
356C3×C4—82no—
357C3×C4—82no—
358C3×C4#53511yescharacteristic
359C3×C4#53611yescharacteristic
360C3×C4#54212yes—
361C23—42no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—42no—
366C23C22×C3×D411yescharacteristic
367Q8—22no—
368Q8—22no—
369Q8—42no—
370Q8—41no—
371Q8—41no—
372Q8—42no—
373Q8—84no—
374D4—41no—
375D4—41no—
376D4—42no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4C3×#12112yes—
400C2×C4C3×#12112yes—
401C2×C4C22×C3×D411yescharacteristic
402C2×C4C2×C3×#4111yescharacteristic
403C2×C4C3×H(5,𝔽2)12yes—
404C2×C4C3×2−1+412yes—
405C8—41no—
406C8—41no—
407C8—42no—
408C2×C3—161no—
409C2×C3#262011yescharacteristic
410C2×C3#304311yescharacteristic
411C2×C3#304611yescharacteristic
412C2×C3#325812yes—
413C2×C3#326312yes—
414C22—81no—
415C22—82no—
416C22—82no—
417C22—81no—
418C22—81no—
419C22C2×C3×#12111yescharacteristic
420C22C3×#53712yes—
421C22C3×#57712yes—
422C22C3×#57711yescharacteristic
423C22C3×#57811yescharacteristic
424C4—22no—
425C4—22no—
426C4—42no—
427C4—42no—
428C4—42no—
429C4—42no—
430C4—81no—
431C4—82no—
432C4—82no—
433C4C3×#53511yescharacteristic
434C4C3×#53611yescharacteristic
435C4C3×#54212yes—
436C3#2984111yescharacteristic, direct product factor, Sylow
437C2—161no—
438C2C3×#262011yescharacteristic
439C2C3×#304311yescharacteristic
440C2C3×#304611yescharacteristic
441C2C3×#325812yes—
442C2C3×#326312yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×#376 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×C3×#475C2semidirect
4. C3×#2320C2semidirect
5. C3×#2066C2semidirect
7. C3×#2056C2semidirect
9. C3×#1987C2semidirect
11. C3×#1843C2semidirect
12. #29841C3direct (N × H)
44. C2×#475C2×C3semidirect
47. #2320C2×C3semidirect
48. #2066C2×C3semidirect
50. #2056C2×C3semidirect
52. #1987C2×C3semidirect
54. #1843C2×C3semidirect
436. C3#29841direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C3×#4752
C2×C3×#4642
C3×#26402
C3×#23202
C3×#20662
C3×#20652
C3×#20562
C3×#20552
C3×#19872
C3×#19832
C3×#18432
#298413